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9.2.3. Example of Linear Expansion
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Today, we will explore linear expansion, which is the change in length of a solid when its temperature changes. How many of you have noticed that when a metal object like a spoon gets hot, it feels longer?
I think I have! Does it actually get longer?
Yes, it does! This change can be measured using the formula ΔL = αL₀ΔT. Can anyone tell me what the symbols in this formula represent?
ΔL is the change in length, right?
Correct! And L₀ represents the original length. How about α?
It’s the coefficient of linear expansion!
Exactly! This coefficient tells us how much a material expands per degree of temperature increase.
What happens if the temperature decreases? Does it shrink back?
Good question! Yes, as the temperature decreases, the material shrinks back to its original length.
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Let's look at an example. Say we have a metal rod 3 meters long at 20°C, and we heat it to 100°C with a coefficient of linear expansion α at 1.5 × 10⁻⁵ °C⁻¹. Who can help me calculate the change in length?
I can! We can use the formula ΔL = αL₀ΔT.
That's right! Substituting the values, what do we get?
ΔL = (1.5 × 10⁻⁵) × 3 × (100 - 20) = 0.0036 m!
Excellent! So the rod expands by 3.6 mm. Always remember, in engineering, we need to consider these changes!
What if the rod was a different material with a different α?
That's a great point! Different materials expand at different rates. Knowing the coefficient is crucial for accurate calculations.
Overview
Short Summary
Linear expansion refers to the change in length of a solid due to a temperature change, described by a specific equation.
Medium Summary
In this section, the concept of linear expansion is introduced, emphasizing its formula, the coefficient of linear expansion, and practical examples demonstrating how it is calculated. It is a fundamental concept in thermal expansion that plays a crucial role in materials engineering and science.
Detailed Summary
Example of Linear Expansion
Linear expansion refers to the phenomenon where the length of a solid increases in response to a rise in temperature. This section delves into the concept using the formula:
ΔL = αL₀ΔT
Where:
- ΔL = Change in length (in meters)
- α = Coefficient of linear expansion (in °C⁻¹)
- L₀ = Original length of the solid (in meters)
- ΔT = Change in temperature (in °C)
The coefficient of linear expansion (α) is a material property that quantifies how much a material expands per degree change in temperature. The example provided calculates the change in length of a metal rod when it is heated from an initial temperature of 20°C to 100°C, illustrating the practical application of the formula.
Understanding linear expansion is essential in engineering fields, where temperature changes in materials must be considered to avoid structural failures.
Reference YouTube Videos
Audio Book
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Create a free accountA metal rod of length 3 m is heated from 20°C to 100°C. If the coefficient of linear expansion is 1.5×10−5 °C⁻¹, the change in length is: ΔL=(1.5×10−5)×3×(100−20)=0.0036 m
Detailed Explanation
In this example, we start with a metal rod that is 3 meters long. When we heat the rod, its temperature rises from 20 degrees Celsius to 100 degrees Celsius. The coefficient of linear expansion, which tells us how much a unit length of the material will expand for every degree of temperature increase, is given as 1.5×10−5 °C⁻¹. To calculate the change in length (ΔL), we can use the formula: ΔL = αL0ΔT, where α is the coefficient of linear expansion, L0 is the initial length, and ΔT is the change in temperature. Substituting the values into the formula gives us ΔL = (1.5×10−5) × 3 × (100 - 20), which results in a change in length of 0.0036 meters, or 3.6 millimeters.
Examples & Analogies
Think of a train track on a hot summer day. As the temperature rises, the metal rail expands, similar to how the metal rod in our example grows longer. If there were no allowance for this expansion, the track could buckle under the pressure, much like if you tried to stretch a rubber band beyond its limit. Understanding how materials expand helps engineers design safe and functional structures.
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Create a free accountHence, the length increases by 3.6 mm.
Detailed Explanation
Once we have calculated the change in length (ΔL) to be 0.0036 meters, we can convert this into millimeters for easier understanding. Since 1 meter equals 1000 millimeters, we multiply the change in length (0.0036 m) by 1000 to get the result in millimeters. This shows that the rod has expanded by 3.6 millimeters.
Examples & Analogies
Imagine a balloon on a sunny day. When the air inside it gets warm, the balloon swells and gets bigger. In this case, the balloon's increase in size symbolizes how our metal rod has lengthened due to heat, with every degree of temperature creating a small increase, adding up to a noticeable change over time.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Linear Expansion: The effect of temperature on the length of a solid.
Coefficient of Linear Expansion (α): Measures how much a material expands per degree increase in temperature.
Application of Linear Expansion Formula: Understanding how to compute length change using the formula ΔL = αL₀ΔT.
Examples
Memory Aids
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Stories
Flash Cards
Glossary
Linear Expansion
The increase in length of a solid when its temperature rises.
Coefficient of Linear Expansion (α)
A material property that indicates the fractional change in length per degree of temperature change.
ΔL
Change in length of the solid.
L₀
Original length of the solid.
ΔT
Change in temperature (in °C).